Pair correlation function closure

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

Here, H and C are symmetric matrices whose elements are the partial total hap(r) and direct cap,(r) pair correlation functions a,ft = A,B) W is the matrix of intramolecular correlation functions wap r) that characterize the conformation of a macromolecule and its sequence distribution and p is the average number density of units in the system. Equation 17 is complemented by the closure relation corresponding to the so-called molecular Percus-... 

The integral equation theory consists in obtaining the pair correlation function g(r) by solving the set of equations formed by (1) the Omstein-Zernike equation (OZ) (21) and (2) a closure relation [76, 80] that involves the effective pair potential weff(r). Once the pair correlation function is obtained, some thermodynamic properties then may be calculated. When the three-body forces are explicitly taken into account, the excess internal energy and the virial pressure, previously defined by Eqs. (4) and (5) have to be, extended respectively [112, 119] so that... 

The equations described earlier contain two unknown functions, h(r) and c(r). Therefore, they are not closed without another equation that relates the two functions. Several approximations have been proposed for the closure relations HNC, PY, MSA, etc. [12]. The HNC closure can be obtained from the diagramatic expansion of the pair correlation functions in terms of density by discarding a set of diagrams called bridge diagrams, which have multifold integrals. It should be noted that the terms kept in the HNC closure relation still include those up to the infinite orders of the density. Alternatively, the relation has been derived from the linear response of a free energy functional to the density fluctuation created by a molecule fixed in the space within the Percus trick. The HNC closure relation reads... 

The results summarised here are for pure water at the temperature 25°C and the density 1.000 g cm , and are obtained by solving numerictdly the Ornstein-Zemike (OZ) equation for the pair correlation functions, using a closure that supplements the hypernet-ted chain (HNC) approximation with a bridge function. The bridge function is determined from computer simulations as described below. The numerical method for solving the OZ equation is described by Ichiye and Haymet and by Duh and Haymet. ... 

The so-called product reactant Ornstein-Zernike approach (PROZA) for these systems was developed by Kalyuzhnyi, Stell, Blum, and others [46-54], The theory is based on Wertheim s multidensity Ornstein-Zernike (WOZ) integral equation formalism [55] and yields the monomer-monomer pair correlation functions, from which the thermodynamic properties of the model fluid can be obtained. Based on the MSA closure an analytical theory has been developed which yields good agreement with computer simulations for short polyelectrolyte chains [44, 56], The theory has been recently compared with experimental data for the osmotic pressure by Zhang and coworkers [57], In the present paper we also show some preliminary results for an extension of this model in which the solvent is now treated explicitly as a separate species. In this first calculation the solvent molecules are modelled as two fused charged hard spheres of unequal radii as shown in Fig. 1 [45],... 

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

As in the elastic case, it will be necessary to provide a closure for the pair correlation functions appearing in Eq. (6.5). 

Owing to the presence of the pair correlation function, the collision model in Eq. (6.2) is unclosed. Thus, in order to close the kinetic equation (Eq. 6.1), we must provide a closure for written in terms of /. The simplest closure is the Boltzmann Stofizahlansatz (Boltzmann, 1872) ... 

The next step is to provide a closure for the pair correlation function appearing in the collision source and collisional-flux terms. For moderately dense flows, the collision frequency for finite-size particles is known to be larger than that found using the Boltzmann Stofizahlansatz (Carnahan Starling, 1969 Enksog, 1921). In order to account for this effect, the pair correlation function can be modeled as the product of two single-particle velocity distribution functions and a radial distribution function ... 

It should be clear that the pair correlation function has, in general, two contributions. One is due to interaction, which in this case is unity. The second arises from the closure condition with respect to N. Placing a particle at a fixed position changes the conditional density of particles everywhere in the system from N/V into (N— )/V. Hence, the pair correlation due to this effect is... 

The structure of the integral equation approach for calculating the angular pair correlation function g(ri2C0iC02) starts with the OZ integral equation [8.76] between the total (h) and the direct (c) correlation function, which is here schematieally rewritten as h=h[c] where h[c] denotes a functional of c. Coupled to that a second relation, the so-ealled closure relation c=c[h], is introduced. While the former is exact, the latter relation is approximated the form of this approximation is the main distinction among the various integral equation theories to be described below. 

In the list of the various correlation functions we have introduced at item (b) the so-called site-site pair correlation functions gap(ro ) since they depend only on the radial variables r p (between sites), they are naturally simpler than g(r,2 co,(02) but, at the same time, they contain less information. The theories for g p fall into two categories based on site-site or particle-particle OZ equations, respectively. Various closures can be used with either category. 

The calculations were performed solving both the HNC2 closure for the inhomogeneous pair correlation function, and also the MSA2 closure in a few cases. A simplified version of the WLMB equation that produces reasonably good results was studied by Colmenares and Olivares [77, 78]. 

Note that the R-MMSA and R-MPY/HTA approximations for the direct correlation functions are now deterministic, that is, uncoupled from the determination of the full pair correlation functions gv,M (r). However, for the most complex R-MPY closure the direct correlation functions are still self-consistently linked to the full radial distribution functions and tail potentials. 

Using the diagrammatic approach, the atomic hypernetted chain (HNC) closure is obtained by neglecting all diagrams which are free of nodal circles (bridge diagrams) in the cluster expansion of the pair correlation function [6], The closure based on HNC theory is shown in Eq. (26). 

Finally, the closure relations for the inhomogeneous pair functions must be chosen. The PY approximation for the fluid-fluid direct correlation function presumes that its blocking part vanishes. This implies that c, ii(/,y) = 0, and... 

From what precedes, it is obvious that in order to determine the latter two correlation functions for a given pair potential u(r), Eq. (21) must be supplemented by an auxiliary closure relation [7, 17, 18, 27] derived from a cluster diagram analysis that reads... 

Assuming the pair potential known, the radial distribution function for two-dimensional systems can be calculated using the two-dimensional version of the Ornstein-Zernike equation, Eq. (22), and one of the closure relations. Although Eq. (22) does not relate one to one the radial distribution function with the pair potential, one might attempt to invert the procedure to get u(r) from the experimental values for g(r). Thus, by taking the Fourier-Bessel (FB) transform [43,44] of Eq. (22) an expression for c(k) is obtained in terms of the FB transform of the measured total correlation function, i.e. 

Then, this equation can be transformed back to the real space to get the experimental direct correlation function c(r). The effective pair potential may now be obtained using a closure relation such as HNC, MSA, or PY. 

Why should one go to all this trouble and do all these integrations if there are other, less complex methods available to theorize about ionic solutions The reason is that the correlation function method is open-ended. The equations by which one goes from the gs to properties are not under suspicion. There are no model assumptions in the experimental determination of the g s. This contrasts with the Debye-Htickel theory (limited by the absence of repulsive forces), with Mayer s theory (no misty closure procedures), and even with MD (with its pair potential used as approximations to reality). The correlation function approach can be also used to test any theory in the future because all theories can be made to give g(r) and thereafter, as shown, the properties of ionic solutions. 

The RSOZ equations Eqs. (7.41) and (7.42) still involve both the total and the direct correlation functions. Therefore, appropriate closure expressions relating the correlation functions to the pair potentials are needed to calculate the correlation functions at given densities and temperatures. Typically, one uses standard closure expreasions familiar from bulk liquid state theory [30]. One should note, however, that the performance of these closures for disordered fluids can clearly not be taken for granted. Instead, they need to be tested for each new model system under consideration. 

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